MIS209 INTRODUCTION TO MANAGEMENT SCIENCE

Transcription

1 MIS209 INTRODUCTION TO MANAGEMENT SCIENCE Introduction to Management Science - Modeling H. Kemal İlter, B.Eng., M.B.A., Ph.D. Assoc. Prof. of Operations D-314, Business School Yildirim Beyazit University mis209@hkilter.com Ankara Fall 2015

2 CLASSIFICATION OF MATHEMATICAL MODELS Purpose of the model Optimization Models Seek to maximize a quantity (profit, efficiency, etc.) or minimize a quantity (cost, time, etc.) that may be restricted by a set of constraints (limitations on the availability of capital, personnel, supplies, etc.) Prediction Models Describe or predict events (sales forecasts, project completion dates, etc.) given certain conditions. Degree of certainty of the data in the model Deterministic models Profit, cost, and resource data are assumed to be known with certainty. Probabilistic or Stochastic models One or more of the input parameters values are determined by probability distributions

3 PAST, PRESENT, FUTURE Management science is generally applied in three situations: 1. Designing and implementing new operations or procedures. 2. Evaluating an ongoing set of operations or procedures. 3. Determining and recommending corrective action for operations and procedures that are producing unsatisfactory results.

4 MANAGEMENT SCIENCE PROCESS A. Problem Definition B. Mathematical Modeling C. Solution of the Model A Problem Definition D. Post-solution Phase B Mathematical Modeling C Solution of the Model D Post-solution Phase

5 MANAGEMENT SCIENCE PROCESS A. Problem Definition 1. Observe operations 2. Ease up on complexity 3. Recognize political realities 4. Decide what is really wanted 5. Identify constraints Create a limiting condition in words in the following manner: The amount of a resource required Has some relation to The availability of the resource 6. Seek conditions feedback Homework 17 Define a real-life business problem. You may use below template to define the problem; Part A 1. What are your observations on the operations of the firm in this situation? 2. Which assumptions do you want to consider in the situation for reducing complexity? 3. Which social or managerial affairs should be considered in this situation? 4. Which part of this situation is create the problem? 5. What are the conditions as identified constraints or restrictions about this problem? 6. How do these conditions affect the problem? Part B Write down your mathematical model to solve this specific problem.

6 MANAGEMENT SCIENCE PROCESS B. Mathematical Modeling 1. Identify decision variables Asking a question; Does the decision maker have the authority to decide the numerical value (amount) of the item? If the answer Yes it is a decision variable. 2. Quantify the objective and constraints Objective function The objective of all optimization models, is to figure out how to do the best you can with what you ve got. The best you can implies maximizing something (profit, efficiency...) or minimizing something (cost, time...). Writing Constraints Make sure the units on the left side of the relation are the same as those on the right side. Translate the words into mathematical notation using known or estimated values for the parameters and the previously defined symbols for the decision variables. Rewrite the constraint, if necessary, so that all terms involving the decision variables are on the left side of the relationship, with only a constant value on the right side. 3. Construct a model shell In the formative stage of model building, generic symbols can be used for the parameters until the actual data are determined.

7 MANAGEMENT SCIENCE PROCESS B. Mathematical Modeling 4. Data gathering The time and cost of collecting, organizing, and sorting relevant data. The time and cost of generating a solution approach; - Make some assumptions, so that a standard solution technique may be used, - Develop a new technique, or modify an existing one. The time and cost of using a model.

8 MANAGEMENT SCIENCE PROCESS C. Solution of the Model 1. Choose an appropriate solution technique 2. Generate model solutions 3. Test/validate model results 4. Return to modelling step if results are unacceptable 5. Perform what-if analyses

9 MANAGEMENT SCIENCE PROCESS D. Post-Solution Phase 1. Prepare a business report or presentation 2. Monitor the progress of the implementation

10 AN EXAMPLE Problem Statement Delta Hardware Stores Delta Hardware Stores is a regional retailer with warehouses in three cities in California: San Jose, Fresno, and Azusa. Each month, Delta restocks its warehouses with its own brand of paint. Delta has its own paint manufacturing plant in Phoenix, Arizona. Asuza

11 AN EXAMPLE Problem Statement Although the plant s production capacity is sometime inefficient to meet monthly demand, a recent feasibility study commissioned by Delta found that it was not cost effective to expand production capacity at this time. To meet demand, Delta subcontracts with a national paint manufacturer to produce paint under the Delta label and deliver it (at a higher cost) to any of its three California warehouses. Given that there is to be no expansion of plant capacity, the problem is to determine a least cost distribution scheme of paint produced at its manufacturing plant and shipments from the subcontractor to meet the demands of its California warehouses. Asuza National Subcontrator

12 AN EXAMPLE Variable Definition Decision maker has no control over demand, production capacities, or unit costs. The decision maker is simply being asked, 1. How much paint should be shipped this month from the plant in Phoenix to San Jose, Fresno, and Asuza? 2. How much extra should be purchased from the subcontractor and sent to each of the three cities to satisfy their orders? San Jose Fresno National Subcontractor Asuza Phoenix

13 AN EXAMPLE Decision Variables Amount of paint shipped this month : from Phoenix to San Jose x 2 : from Phoenix to Fresno x 3 : from Phoenix to Azusa Amount of paint subcontracted this month x 4 : for San Jose x 5 : for Fresno x 6 : for Azusa San Jose Fresno National Subcontractor Asuza Phoenix

14 AN EXAMPLE Model Shell The objective is to minimize the total overall monthly costs of manufacturing, transporting and subcontracting paint, subject to: 1. The Phoenix plant cannot operate beyond its capacity. 2. The amount ordered from subcontractor cannot exceed a maximum limit. 3. The orders for paint at each warehouse will be fulfilled. To determine the overall costs; m : The manufacturing cost per 1000 gallons of paint at the plant in Phoenix c : The procurement cost per 1000 gallons of paint from National Subcontractor t 1, t 2, t 3 : The respective truckload shipping costs form Phoenix to San Jose, Fresno, and Azusa s 1, s 2, s 3 : The fixed purchase cost per 1000 gallons from the subcontractor to San Jose, Fresno, and Azusa To write to constraints, we need to know; q 1 : The capacity of the Phoenix plant q 2 : The maximum number of gallons available from the subcontractor r 1, r 2, r 3 : The respective orders for paint at the warehouses in San Jose, Fresno, and Azusa

15 AN EXAMPLE Objective Function and Constraints Objective Function Minimize Z = Total Cost or Minimize Z = Cost Manufacturing + Cost Transporting + Cost Subcontracting + Cost Purchasing or Minimize Z = m + m x 2 + m x 3 + t 1 + t 2 x 2 +t 3 x 3 + c x 4 + c x 5 + c x 6 + s 1 x 4 + s 2 x 5 +s 3 x 6 or Minimize Z = (m + t 1 ) + (m + t 2 ) x 2 + (m + t 3 ) x 3 + (c + s 1 ) x 4 + (c + s 2 ) x 5 + (c + s 3 ) x 6 Constraints The number of truckloads shipped out from Phoenix cannot exceed the plant capacity: + x 2 + x 3 q 1 The number of thousands of gallons ordered from the subcontrator cannot exceed the order limit: x 4 + x 5 + x 6 q 2 The number of thousands of gallons received at each warehouse equals the total orders of the warehouse: + x 4 = r 1 x 2 + x 5 = r 2 x 3 + x 6 = r 3 All shipments must be nonnegative and integer:, x 2, x 3, x 4, x 5, x 6 0, x 2, x 3, x 4, x 5, x 6 integer

16 AN EXAMPLE The General Model and Data Collection General mathematical model (shell) for the example Minimize Z = (m + t 1 ) + (m + t 2 ) x 2 + (m + t 3 ) x 3 + (c + s 1 ) x 4 + (c + s 2 ) x 5 + (c + s 3 ) x 6 Subject to: + x 2 + x 3 q 1 x 4 + x 5 + x 6 q 2 + x 4 = r 1 x 2 + x 5 = r 2 x 3 + x 6 = r 3, x 2, x 3, x 4, x 5, x 6 0, x 2, x 3, x 4, x 5, x 6 integer Data Collection Respective orders (gallons) r 1 = 4,000, r 2 = 2,000, r 3 = 5,000 Capacity (gallons) q 1 = 8,000, q 2 = 5,000 Subcontractor price, c = $5,000 per 1,000 gallons Cost of production, m = $3,000 per 1,000 gallons Transportation costs, Subcontractor, s 1 = $1,200, s 2 = $1,400, s 3 = $1,100 Phoenix Plant, t 1 = $1,050, t 2 = $750, t 3 = $650

17 AN EXAMPLE The Model for this Month and the Solution Mathematical model for this month Minimize Z = 4, ,750 x 2 + 3,650 x 3 + 6,200 x 4 + 6,400 x 5 + 6,100 x 6 Subject to: + x 2 + x 3 8,000 x 4 + x 5 + x 6 5,000 + x 4 = 4,000 x 2 + x 5 = 2,000 x 3 + x 6 = 5,000, x 2, x 3, x 4, x 5, x 6 0, x 2, x 3, x 4, x 5, x 6 integer Results = 1,000 gallons x 2 = 2,000 gallons x 3 = 5,000 gallons x 4 = 3,000 gallons x 5 = 0 x 6 = 0 Total Cost = $44,800

18 BREAK-EVEN ANALYSIS Profit = Income Outcome Income = Selling price per unit Number of units Outcome = Fixed cost + (Variable cost per unit Number of units) Revenue and Cost ($) Income Total Cost a Break-Even Point Variable Cost Fixed Cost n Production volume

19 ADDENDUM Homework 23 Three types of manufacturing equipment for engine gaskets are under consideration. Their fixed costs and resulting variable costs per unit are shown in the table below. Fixed cost ($) Variable cost ($)/unit Equipment A 4, Equipment B 7, Equipment C 12, a. At what volume of production would Equipment A and Equipment B cost the same? b. Establish this breakeven point for equipments B and C. c. Suppose the volume anticipated was 8,000 units. Which equipment should be purchased? d. Suppose the volume anticipated was 12,900 units. Which equipment should then be purchased? Homework 47 Two brothers want to open a small neighborhood bakery, where they will specialize in loaves of bread. They have carefully estimated their fixed costs to be $110,000 per year. This includes a salary of $35,000 for each brother. The facility they will use is a former pizza kitchen. One problem is that there is limited capacity, so there will be an upper limit on the number of loaves of bread they can produce. Baking 6 days a week, the brothers believe they can produce 150 loaves per day. The cost to them for each loaf is $0.55. They believe they will be able to sell all the bread they can bake. They plan to bake 312 days during the year. a. Given these projections, how much will the brothers have to charge per loaf to stay in business? b. Suppose they are willing to take a much lower salary to get their business started. If they each agreed to take only $25,000 per year, how much would they have to charge for their bread?

20 ADDENDUM Homework 793 The ThinkBig Company currently manufactures a product which sells for $1.30. The fixed costs associated with this operation are $18,000; the variable costs are $0.65 per unit on a volume of 35,000 units per month. They are considering new equipment which will increase the fixed costs to $26,000 and the variable costs to $0.75, but the demand is expected to increase to 55,000 units, so this option may be attractive. Should ThinkBig make this investment? Why or Why not? The ThinkBig Company is now ready to purchase the new equipment. Two changes to the above plan include a price change to $1.45 per unit, and a revised demand forecast to 45,000 units. Given these circumstances, should the investment be made? Problem from the Textbook George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund. Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at least 30% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives.

21 ADDENDUM Problem from the Textbook The Sea Wharf Restaurant would like to determine the best way to allocate a monthly advertising budget of $1000 between newspaper advertising and radio advertising. Management decided that at least 25% of the budget must be spent on each type of media, and that the amount of money spent on local newspaper advertising must be at least twice the amount spent on radio advertising. A marketing consultant developed an index that measures audience exposure per dollar of advertising on a scale from 0 to 100, with higher values implying greater audience exposure. If the value of the index for local newspaper advertising is 50 and the value of the index for spot radio advertising is 80, how should the restaurant allocate its advertising budget in order to maximize the value of total audience exposure? Formulate a linear programming model that can be used to determine how the restaurant should allocate its advertising budget in order to maximize the value of total audience exposure. Problem from the Textbook Tom s, Inc., produces various Mexican food products and sells them to Western Foods, a chain of grocery stores located in Texas and New Mexico. Tom s, Inc., makes two salsa products: Western Foods Salsa and Mexico City Salsa. Essentially, the two products have different blends of whole tomatoes, tomato sauce, and tomato paste. The Western Foods Salsa is a blend of 50% whole tomatoes, 30% tomato sauce, and 20% tomato paste. The Mexico City Salsa, which has a thicker and chunkier consistency, consists of 70% whole tomatoes, 10% tomato sauce, and 20% tomato paste. Each jar of salsa produced weighs 10 ounces. For the current production period, Tom s, Inc., can purchase up to 280 pounds of whole tomatoes, 130 pounds of tomato sauce, and 100 pounds of tomato paste; the price per pound for these ingredients is $0.96, $0.64, and $0.56, respectively. The cost of the spices and the other ingredients is approximately $0.10 per jar. Tom s, Inc., buys empty glass jars for $0.02 each, and labeling and filling costs are estimated to be $0.03 for each jar of salsa produced. Tom s contract with Western Foods results in sales revenue of $1.64 for each jar of Western Foods Salsa and $1.93 for each jar of Mexico City Salsa. Develop a linear programming model that will enable Tom s to determine the mix of salsa products that will maximize the total profit contribution.